Note that the gain of the ``current input sample'' is now fixed at
for all time. Also note that we can handle only length 3 FIR
filters in this representation, and that the output signal is ``cut
off'' at time
. The cut-off time is one sample after the filter
is fully ``engaged'' by the input signal (all filter coefficients see
data). Even if the input signal is zero at time
and beyond, the
filter should be allowed to ``ring'' for another two samples. We can
accommodate this by appending two zeros to the input and going with a
banded Toeplitz filter matrix:

(F.3)

We could add more rows
to obtain more output samples,
but the additional outputs would all be zero.

In general, if a causal FIR filter is length
, then its order is
, so to avoid ``cutting off'' the output signal prematurely, we
must append at least
zeros to the input signal. Appending
zeros in this way is often called zero padding, and it is used
extensively in spectrum analysis [84]. As a specific example,
an order 5 causal FIR filter (length 6) requires 5 samples of
zero-padding on the input signal to avoid output truncation.

If the FIR filter is noncausal, then zero-padding is needed
before the input signal in order not to ``cut off'' the
``pre-ring'' of the filter (the response before time
).

To handle arbitrary-length input signals, keeping the filter length at
3 (an order 2 FIR filter), we may simply use a longer banded Toeplitz
filter matrix:

A complete matrix representation of an LTI digital filter (allowing
for infinitely long input/output signals) requires an infinite
Toeplitx matrix, as indicated above. Instead of working with infinite
matrices, however, it is more customary to speak in terms of
linear operators [56]. Thus, we may say
that every LTI filter corresponds to a Toeplitz linear
operator.